Almost sure convergence coding theorems of one-shot and multi-shot Tunstall codes are proved for stationary memoryless sources. Coding theorem of one-shot Tunstall code is proved in the case that the leaf count of Tunstall tree increases. On the other hand, coding theorem is proved for multi-shot Tunstall code with increasing parsing count, under the assumption that the Tunstall tree grows as the parsing proceeds. In this result, it is clarified that the theorem for the one-shot Tunstall code is not a corollary of the theorem for the multi-shot Tunstall code. In the case of the multi-shot Tunstall code, it can be regarded that the coding theorem is proved for the sequential algorithm such that parsing and coding are processed repeatedly. Cartesian concatenation of trees and geometric mean of the leaf counts of trees are newly introduced, which play crucial roles in the analyses of multi-shot Tunstall code.